Generating sets for ideals of finite type in \(H^\infty\)
type: Item
instance of: scholarly article
title: Generating sets for ideals of finite type in \(H^\infty\)
zbMATH Open document ID: 1257.46023
author: Raymond Mortini
published in: Bulletin des Sciences Mathématiques
publication date: 2012-09-19T00:00:00Z
Mathematics Subject Classification ID: 46J15
Mathematics Subject Classification ID: 46J20
Mathematics Subject Classification ID: 30H05
Mathematics Subject Classification ID: 30H80
Mathematics Subject Classification ID: 30J10
zbMATH DE Number: 6084110
zbMATH Keywords: algebra
zbMATH Keywords: ideal
zbMATH Keywords: bounded analytic functions
zbMATH Keywords: Blaschke product
zbMATH Keywords: generalized corona theorems
zbMATH Keywords: finitely generated ideals
reviewed by: Anthony O'Farrell
MaRDI profile type: Publication
full work available at URL: j.bulsci.2012.02.002
OpenAlex ID: W2163387443
cites work: Interpolations by bounded analytic functions and the corona problem
cites work: Q3922907
cites work: Q4869349
cites work: Two new characterizations of Carleson-Newman Blaschke products
cites work: Higher order hulls in \(H^\infty\). II
cites work: The generalized corona theorem
cites work: Bounded analytic functions and Gleason parts
cites work: Q3881402
cites work: Discrete Carleson measures and some interpolation problems
cites work: Coherence of some rings of functions
cites work: IDEALS GENERATED BY INTERPOLATING BLASCHKE PRODUCTS
cites work: Finitely generated ideals in the disk algebra
cites work: On an example of J. Bourgain on closures of finitely generated ideals
cites work: Q3713006
cites work: On a generalized corona problem on the unit disc
cites work: Q5484158
cites work: Finitely generated ideals in the Banach algebra \(H^\infty\)
cites work: Interpolating Blaschke products and the ideals of the algebra \(H^{\infty}\)
cites work: Blaschke products satisfying the Carleson-Newman conditions and ideals of the algebra \(H^{\infty}\)
cites work: Estimates in the corona theorem and ideals of \(H^\infty\): a problem of T. Wolff.
cites work: The problem of ideals of \(H^{\infty }\): beyond the exponent 3/2
DOI: 10.1016/J.BULSCI.2012.02.002
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review text: In the algebra \(H^\infty\) of bounded analytic functions in the unit disc \(\mathbb D\), let \(I(f_1,\dots,f_n)\) denote the ideal generated by the elements \(f_1,\dots,f_n\), and let \(J(f_1,\dots,f_n)\) denote the a priori larger ideal NEWLINE\[NEWLINE \left\{ f\in H^\infty: \exists C>0 \text{ such that } |f|\leq C\sum_{j=1}^n |f_j| \right\} NEWLINE\]NEWLINE -- a so-called ideal of finite type.NEWLINENEWLINEThis well-written paper is mainly concerned with the conditions under which \(I=J\). The author previously [Bull. Aust. Math. Soc. 50, No. 3, 521--528 (1994; Zbl 0823.46051)] settled the corresponding question in the disc algebra. There is also previous work on the question by von Renteln [\textit{M. von Renteln}, Collect. Math. 26, 115--126 (1975; Zbl 0315.46044)] and Tolokonnikov [\textit{V. A. Tolokonnikov}, J. Sov. Math. 27, 2549--2553 (1984; Zbl 0546.46046)]. Previous work is reviewed, and the paper proceeds to assemble a large number of partial results that have been worked out by the author, in part with P. Gorkin, over a 20 year period. The problem is connected to the trivial Gleason parts of \(H^\infty\), to the number of generators of an order \(N\) ideal, and the techniques include Carleson measures and Carleson-Newman Blaschke products (finite products of interpolating Blaschke products), and higher-order zero sets. A sample result (of 22 in all) is Theorem 4.3: Let \(b\) and \(c\) be two Carleson-Newman Blaschke products of order 2. Then the ideal \(J(b,c)\) is three-generated. Additionally, if \(b\) and \(c\) have no common zeros in \(\mathbb D\), then there exist interpolating Blaschke products \(B\), \(B^*\), \(C\) and \(C^*\) such thatNEWLINENEWLINE1. \(b=BB^*\), \(c=CC^*\),NEWLINENEWLINE 2. \(J(b,c)=I(BB^*,CC^*,BC)\).NEWLINENEWLINE In particular, we have \(J(BB^*,CC^*,BC)=I(BB^*,CC^*,BC)\).NEWLINENEWLINEThe paper also includes results about intersections of finitely-generated ideals, and a number of open problems are posed.
description: scientific article; zbMATH DE number 6084110